Appendix C
A General Fibonacci Calculation:
The Fibonacci infinite sequence was referenced in Appendix A,
F(n) = F(n-1) + F(n-2) with seed values F(0) = 0 and F(1) = 1.
Ratios converge, and
lim(n infinity) F(n+1) / F(n) = φ = (1 + 5^1/2) / 2 = .618 … and
lim(n infinity) F(n-2) / F(n) = γ = .382 … and so on.
Writing an example expression for spatial dimension ≥ 3 per Appendix A
∫∫∫∫∫∫∫∫dV = ∫∫∫∫∫dV(0) x exp(rV x t)
where rV = r-sub-V = φ^(D(n+1) – D(n))
then
dx /dx(0) = exp(φ ^ (D(n+1) – D(n))^1/(n+1).
Except we are now doing math in another dimension, and while e = 2.718 in three dimensions, the base of natural logarithms should change in higher or lower dimensional space.
For example, in the case of 5 dimensions: e e^1 / γ^1.
We quickly find dx /dx(0) = 1.08.
A different example, for the case of spatial dimension < 3:
The base e must change as a function of the power of B, i.e. in three dimensional space B ~ sec^2 while in two-dimensional space B ~ sec^3/2.
The difference in power of physical events B
2 – 3/2 = 1/2 and the two-dimensional e = 2.718^1/2.
Then we quickly find dx /dx(0) = 1.08 similar to the previous mean calculations for the difference between b(min) and b-empirical.
The (Fibonacci) calculations hold true for any spatial dimension n moving through n+1 with a dimensional adjustment for e.
